Definition: (Binary Relation) A Binary Relation on is a subset on , if , we write and .
Definition: (Reflexivity) A relationship is reflexive if .
Definition: (Transitivity) A relationship is transitive if implies .
Definition: (Completeness) A relationship is complete if either or .
Definition: (Symmetry) A relationship is symmetric if implies .
Definition: (Anti-Symmetry) A relationship is anti-symmetric if and implies .
Definition: (Preorder) is a Preorder if is reflexive and transitive.
Definition: (Partial Order) R is a Partial Order if is transitive and antisymmetric.
Definition: (Linear Order) R is a Linear Order if is a complete partial order.
Assumption: (Consumption) The consumer has preferences over consumption bundle , which is a nonempty, convex, compact subset of . is always in .
Definition: (Preference) A Preference relation is a binary relation on the set of alternatives , denoted as .
Definition: (Strict Preference) A Strict Preference relation is a binary relation on the set of alternatives , denoted as , if but .
Definition: (Indifferent Preference) An Indifferent Preference relation is a binary relation on the set of alternatives , denoted as , if but .
Definition: (At Least As Good Set) At Least As Good Set is defined as .
Definition: (No Better Than Set) No Better Than Set is defined as .
Definition: (Strictly Preferred To Set) Strictly Preferred To Set is defined as .
Definition: (Strictly Worse Set) Strictly Worse Set is defined as .
Definition: (Indifferent Curve) Indifferent Curve is defined as .
Axiom: (A1 - Completeness) For any distinct bundles and , either or .
Axiom: (A2 - Transitivity) For any distinct bundles and , implies .
Lemma: (Completeness and Transitivity) if is complete and transitive, then and are also complete and transitive.
Definition: (Rationality) A preference is called rational if it satisfies completeness and transitivity.
Theorem: (Ranking) Given a finite set of bundles , if the preference relation satisfies A1 and A2, then bundles in A can be ranked in a list consistent with the preference.
Proof:
It suffices to show that there is a maximum of . Define an algorithm to find this item. Start with the first item in the set. Suppose next item is preferred to the item in hand, switch the items, i.e. if but do the switching. Keep doing this until we compare every item in the set.
The only potential trouble happens when we switch the items. We want to show that is preferred to any other item before the switching. Suppose there is no switching before. This implies that for . From the switching we know that , by Transitivity, we have for . Now suppose there are numbers of switching happened before. We have for between and , plus we have . This implies that for any . If we repeat this algorithm, we will have for any , i.e. is the maximum.
Axiom: (A3 - Continuity) For any bundle , the set and are closed, or the set and are open.
Theorem: (Continuity) Suppose a preference is complete, transitive and continuous. Then for any , and , there is a number such that .
Proof:
Define a subset of real number: . Defined . Claim that . Suppose , then either or . Suppose , then since is open, there is a such that , contradicting to the fact that is the infimum. Suppose , then since is open, there is a such that , again contradicting to the fact that is the infimum.
Axiom: (A4’ - Local Non-Satiation) For any , for any , there is another bundle such that .
Axiom: (A4 - Strict Monotonicity) For any and , if then , and if if then .
Claim: (Local Non-Satiation and Strictly Monotonicity) If a preference is strictly monotonic then it is locally non-satiated.
Axiom: (A5’ - Convexity) If , then for any .
Axiom: (A5 - Strict Convexity) If , and , then for any .
Axiom: (A6 - Homothetic) If , then for any .
Definition: (Utility Function) A function is called a utility function representing the preference if for any , if and only if .
Theorem: (Existence of Utility Function) If the preference is complete, transitive, continuous, and strictly monatomic, then there is a real-valued and continuous function representing the preference.
Proof:
Let be a vector in . For any , define so that . We claim that such a number is well defined.
Now we will show that represents the preference, i.e. if and only if . We have , and the statement is true as a result of strictly monotonicity.
The last step is to show is continuous. It suffices to show that is an open set. . By continuity, we have both sets are opens, then the intersection of them is also open.
Definition: (Marginal Utility) The marginal utility is defined as .
Definition: (Marginal Rate of Substitution) The Marginal Rate of Substitution is defined as .
Definition: (Elasticity of Substitution) The Elasticity of Substitution is defined as .
Lemma: (Utility and Preference) if and only if and if and only if .
Theorem: (Monotonic Transformation) If is a function that represents the preference, and is another function representing the same preference, then for some strictly increasing function . Inversely, if for some strictly increasing function , then is another function representing the same preference.
Proof:
is equivalent to . Hence the statement is true.
Theorem: (Properties of Utility Function) The following statements are true:
Proof:
These statements are true by definitions. Proof is omitted.
Definition: (Homogeneity) A utility function is called to be homogenous of degree if .
Theorem: (Homogeneity and Homothetic Preference) A utility function is homogenous if and only if the preference is homothetic.
Proof:
Suppose a utility function is homogenous, then . We have if and only if . Then , i.e. , the preference is homothetic.
Suppose the preference is homothetic, then implies . Define a utility function as , where . Then we have . But homothetic preference implies that . By transitivity we have , i.e. is homogenous of degree 1.
Assumption: (Consumer) The utility function is continuous, strictly increasing, strictly quasi-concave.
Definition: (Budget Set) Denote as the set of feasible choices for the consumer, it is a budget set when . Note that there can be goods.
Definition: (Consumer’s Problem) The consumer is solving the Utility Maximizing Problem:
Theorem: (Existence and Uniqueness of the Solution) The solution to the consumer’s problem exists and is unique.
Proof:
The utility function is continuous, the budget set is compact, hence there is a maximum on the constraint.
Now we prove the uniqueness. Suppose there are two solutions to the problem, and . Then and , which implies that . Since both are solutions, we have . Then , contradicting to the fact that is strictly quasi-concave.
Theorem: (Exhausting the Budget) With out the loss of generality, we could impose the budget constraint with equality.
Proof:
Suppose not, then the solution to the problem satisfies , which cannot be true when is strictly increasing.
Solution: (Kuhn-Tucker Method)
Define the Lagrange Function as . The Kuhn-Tucker condition to solve the problem is the first order conditions, i.e.
Theorem: (Kuhn-Tucker Theorem) Suppose is strictly quasi-concave and differentiable, , then if solve the first order conditions, then solve the consumer’s problem.
Proof:
First notice one fact: when , , and , then .
Now prove the statement. Suppose not, the there is another solution such that . By the fact above, we have , and . Notice that since solve the Kuhn-Tucker conditions, we have . By assumption, , so . So implies that , which is a contradiction to the fact that is feasible.
Definition: (Marshallian Demand) The solution the the utility maximizing problem is called the Marshallian Demand, denoted by .
Definition: (Elasticity) The Price Elasticity of the demand is defined as .
Definition: (Indirect Utility Function) The Indirect Utility Function, also known as the Value Function, is defined as .
Theorem: (Properties of the Indirect Utility Function) If is strictly increasing on , then is:
Proof:
The continuity of the value function follows from the theorem of maximum.
The value function is homogenous of degree 0 since is identical to .
consider , suppose and are the solutions to the maximizing problem separately, then . So consider and , but since the utility function is strictly increasing, , hence .
consider , suppose and are the solutions to the maximizing problem separately, then and . Then we have .
Now we want to show that is quasi-convex in . It suffices to show that , where . We will argue by contradiction. This means that there exists , such that , and . This implies that and , then and . Now if we combine them we would get , contradicting to the fact that .
The first proof follows from the Envelope Theorem. We have and , which will give us the Roy’s Identity.
Another proof defines . Consider as a function of prices. Note that for any because is always affordable. This function is minimized at the price . Since the function is differentiable, we have the first order condition: , which will give us the Roy’s Identity.
Definition: (Expenditure Minimizing Problem) The expenditure minimizing problem is defined as
Theorem: (Existence of the solution) Let . If , and is continuous, then there is a solution to the minimization problem. Furthermore, if is strictly increasing and continuous and strictly quasi-concave, and , then the solution is unique.
Proof:
First, if , by definition there is a such that . So is enough to achieve the utility level. So the problem is equivalent to
Now we are minimizing a continuous function over a compact set, hence the solution exists.
Next we prove the uniqueness. Suppose there are two solutions to the problem, we have and . Since and , we have due to the strict quasi-concavity of the utility function. Note that . Then there exists such that and , contradicting to the fact that and are both solutions.
Solution: (Lagrange Method)
Define the Lagrange function as . Take the first order conditions we will get and .
Definition: (Hicks Demand) The solution the the expenditure minimizing problem is called the Hicks Demand, denoted by .
Definition: (Expenditure Function) The Expenditure Function is defined as .
Theorem: (Properties of the Expenditure Function) If is continuous, strictly increasing and strictly quasi-concave, then the expenditure function is:
Proof:
The bundle giving the lowest utility level is . the expenditure to achieve this bundle is trivially zero.
Given the objective function is continuous and the feasible set is compact, by the Theorem of Maximum the expenditure function is continuous.
First prove the monotonicity. Suppose , let , , since is strictly increasing and , we have , since is minimizing the objective function. Now we only need to show that . Suppose , since , we have . There exists some such that because and is continuous. So is feasible for the minimization problem, and , i.e. the expenditure function is strictly increasing in .
Now we prove the unboundedness. Since is increasing and continuous, the attainable utility level is an interval: , where is a finite number or infinity. Pick an increasing sequence of utility level , we want to show that is unbounded above. Suppose this is not true. Since is strictly increasing, let , if is bounded, then is also bounded. Then there is a subsequence of ,denoted as which goes to some . Then since is continuous, we have . Since there is only one limit of the sequence , which is , we have . But then would be attainable, which is impossible.
Suppose , and and are the solutions to the minimizing problem, separately. Since is always attainable, we have .
At price , the problem can be rewrite as , which is identical to the original problem. So the solution is the same, i.e. .
Consider and , define . Let , , and . Then we have and . Combine them we have , i.e. the expenditure function is concave.
The first proof follows from the Envelope Theorem. We have .
An alternative proof is to consider the function , where . This function is maximized at . So the first order condition is .
Lemma: (Duality) Generally, we have , and .
Theorem: (Duality Theorem) If is continuous and strictly increasing, then and .
Proof:
Theorem: (Duality of the Solutions) Suppose is continuous, strictly increasing and strictly quasi-concave. Then we have and for all and U attainable.
Proof:
Theorem: (Slutsky Equation) We have for any .
Proof:
From the duality theorem, we have . Take total differentiation with respect to , we have .
Definition: (Substitution Effect) is defined as the Substitution Effect.
Definition: (Income Effect) is defined as Income Effect.
Definition: (Normal Good) A good is Normal at a point if .
Definition: (Inferior Good) A good is Inferior at a point if .
Definition: (Giffen Good) A good is Giffen at a point if .
Theorem: (Properties of Marshallian Demand) Under all assumptions of the utility function, the Marshallian demand is homogenous of degree 0, and it satisfies the budget balance.
Proof:
We have , which is equivalent to . Since is continuous and strictly increasing, . The budget balance follows from the strictly increasing property of the utility. Suppose , then we can find another bundle such that and , contradicting to the fact that is the solution.
Theorem: (Properties of Hicks Demand) Under all assumptions of the utility function, and suppose , the Hicks demand is homogenous of degree 0 with respect to , and exhausts the utility constraint.
Proof:
At price , the problem can be rewrite as , which is identical to the original problem. So the solution is the same.
Exhausting the utility constraint follows from the strictly increasing property of the utility. Suppose , then we can find another bundle such that and , when . This is contradicting to the fact that is the solution.
Definition: (Slutsky Matrix) Define the Slutsky Matrix as .
Theorem: (Properties of Slutsky Matrix) The Slutsky matrix is symmetric and negative semi-definite.
Proof:
Note that by Slutsky Equation, . By Shephard’s lemma, we have , hence , by Young’s theorem this is symmetric. Since is concave, is negative semi-definite.
Theorem: (Recovering Expenditure Function) Suppose satisfies the seven properties of the expenditure function, then define . is well defined, unbounded, increasing and quasi-concave.
Proof:
Theorem: (Verification) Suppose is any function that satisfies the seven properties of the expenditure function, then define . Then .
Proof:
First we prove that .
Fix and , for any such that , by the definition of , we have for all . This implies . Furthermore, since is increasing in , and , we have . Recall this inequality holds for all such that . Suppose the solution to minimization problem is , we have , i.e. .
Now we prove that .
It suffices to find such that and . Let . We first prove that . By assumption we have that is homogenous of degree 1 in prices. So by Euler’s theorem, . Also, we know that is concave in , so . Plug in the Euler’s theorem, we have for any . By definition, .
Now we prove that . This is true because also satisfies the Shepherd’s lemma, which gives us . Combine 1 and 2, we prove the theorem.
Solution: (Algorithm of Recovering Expenditure Function)
Solve for the recovered expenditure function with the following algorithm:
Theorem: (Homogeneity) If satisfies budget balance and symmetry of associated Slutsky matrix , then it is homogenous of degree 0 in .
Proof:
By budget balance, . . Differentiate with respect to , we will get , or . Now differentiate with respect to , we will get . Now define . We want to show that , i.e. . We have:
Using symmetry, we have
This finishes the proof.
Theorem: (Slutsky Property) Suppose that satisfies budget balance and homogeneity. Then for all , .
Proof:
Note that .
Theorem: (Integrability Theorem) If a differentiable function satisfies budget balance, symmetry and negative semi-definite of Slutsky matrix, then it is the demand function generated by some increasing, quasi-concave function.
Proof:
First we want to show that suppose is an expenditure function that is generated by some utility function, that is not related to the given function . However, when we have for all , the Marshallian demand generated by the same utility function satisfies . This follows from duality and Shepherd’s lemma. We have . For each fixed , as varies assumes all numbers on , so we could change to and have the aimed equation.
Second we want to show that the solution to for all always exists. By Fubini theorem, if is symmetric, the solution exists, which is guaranteed by the assumption of .
Now we want to show that the solution satisfies the 7 properties of expenditure functions.
It is obviously zero when take the lowest utility level. Since we generated the function from a differential equation it is also continuous. By construction the function is strictly increasing and unbounded in since we can always choose the function to be that way. It is increasing in because . It is concave in since is negative semi-definite. Shepherd’s lemma is automatically true. We only need to show that is homogenous of degree 1 in . Since we have , by Euler’s theorem, is homogenous of degree 1.
Definition: (Weak Axiom of Revealed Preference) Consumer choice satisfies the Weak Axiom of Revealed Preference (WARP) if for any pair of , is chosen at price and is chosen at , then implies .
Definition: (Strong Axiom of Revealed Preference) Consumer choice satisfies the Strong Axiom of Revealed Preference (SARP) if for any , is chosen at price and is chosen at , then , and so on implies .
Theorem: (Homogeneity) Suppose the choice function satisfies the budget balance and WARP, then it is homogenous of degree 0.
Proof:
Let and . Suppose . Assuming budget balance and WARP, we have and . since is a number, we have , i.e. is affordable under , by WARP, we have . Similarly since is affordable under , by WARP we have , which is a contradiction.
Theorem: (Slutsky Property) Suppose the choice function satisfies the budget balance and WARP, then its Slutsky matrix is negative semi-definite.
Proof:
Let and consider . for an arbitrary price level , let . Budget balance implies hat . is affordable under , if , by WARP, , and when , we have . Now combine the conditions we have
Let , where and is a vector in . so this becomes . Choose close to zero, we have , i.e. . If we define a function , this implies that the function attains its maximum when . Note that , so the equivalent condition is . So we have
So the Slutsky matrix is negative semi-definite.
Theorem: (Recover WARP) Suppose that a choice function x(p,y) satisfies homogeneity and budget balance. Suppose further that whenever is not proportional to , we have . Then satisfies WARP.
Proof:
Theorem: (Slutsky Property with Two Goods) Suppose there are only two goods and that a consumer’s choice function satisfies budget balance. If is homogeneous of degree zero in , then the Slutsky matrix associated with is symmetric.
Proof:
We want to show that . From budget balance we have . Take total differentiation with respect to , we have . Take total differentiation with respect to , we have . Take total differentiation with respect to , we have . Combine them we have
Now since is homogeneous of degree zero in , Euler’s theorem gives us . Plug this into the equation, we have
And this finishes the proof.
Theorem: (Equivalent Definition of WARP) Suppose that the choice function is homogeneous of degree zero in .Show that WARP is satisfied for all if and only if it is satisfied on .
Proof:
Definition: (Simple Lottery) A simple lottery is a lottery all of whose outcomes are members of finite outcome set . A simple lottery is denoted as .
Axiom: (G1 - Completeness) For any distinct bundles and , either or .
Axiom: (G2 - Reflexivity) .
Axiom: (G3 - Transitivity) For any distinct bundles , and , if then .
Lemma: (Ranking) By G1-G3 we can rank any finite elements in .
Axiom: (G4’ - Archimedean) For any , if , then there exist such that , and .
Axiom: (G4 - Continuity) For any bundles , there exists a probability such that .
Claim: (Archimedean and Continuity) If a preference is continuous, then it is Archimedean.
Axiom: (G5 - Monotonicity) For any and , if and only if .
Axiom: (G6 - Substitution) If and , and for all , then .
Axiom: (G7 - Reduced to simple gamble) If is a compound lottery and is the simple lottery induced by , then .
Axiom: (G8 - Independence) For any and , we have is equivalent to .
Theorem: (Independence Property) If a preference is substitutable and able to reduce to simple gamble, then it is independent.
Proof:
By substitution axiom we have and . By the ability to reduce to simple gamble , we have . Hence by transitivity and , we have .
Definition: (Expected Utility Property) The utility function satisfies the Expected Utility Property if for all , , where is the simple lottery induced by . This is also called a Von-Neumann-Morgenstern Utility Function.
Theorem: (Von-Neumann-Morgenstern Utility Function) If a preference satisfies the axiom G1-G7, then there is a Von-Neumann-Morgenstern utility function representing the preference.
Proof:
By construction, for each gamble , let defined as . Since the preference is continuous, always exists. Since the preference is monotonic, is unique. Suppose , by transitivity we have . By monotonicity we have . The inverse is also true, so represents the preference.
Now we want to show that has the Von-Neumann-Morgenstern property. Let be arbitrary, denote as the simple gamble induced by . By the ability to reduce to simple gamble, we have , or . Notice that for all , so by substitution, we have . Now reduce that to simple gamble, we have , by definition and transitivity, we have .
Theorem: (Unique up to Linear Transformation) Suppose is a Von-Neumann-Morgenstern utility function representing a preference. Then is another Von-Neumann-Morgenstern utility function representing the same preference if and only if .
Proof:
Definition: (Risk Averse) Given a gamble distribution , is the distribution that will yield the expected value of for sure. Then the preference relation is Risk Averse if for all , . It is called Risk Loving if for all , . It is Risk Neutral if for all , .
Note: A preference can be neither risk averse, risk loving nor risk neutral.
Definition: (Certainty Equivalent) Given a strictly increasing and continuous VNM utility function over wealth, the Certainty Equivalent is defined as . Or, is the amount of wealth such that .
Definition: (Risk Premium) Define Risk Premium as .
Definition: (Probability Premium) The Probability Premium is defined as the probability higher than 0.5 such that .
Theorem: (Risk Aversion) Suppose a preference has the expected utility property, the following are equivalent:
Proof:
Definition: (Comparing Risk Aversion) Given 2 preferences, preference 1 is more risk averse than preference 2 if and only if implies .
Definition: (Arrow-Pratt Measure of Absolute Risk Aversion) The Absolute Risk Aversion is defined as for all .
Definition: (Arrow-Pratt Measure of Relative Risk Aversion) The Relative Risk Aversion is defined as for all .
Theorem: (Relative Risk Aversion) Suppose 2 preferences have the expected utility property, the following are equivalent:
Proof:
Definition: (First Order Stochastically Domination) First Order Stochastically Dominate , denoted by , if for all non-decreasing function .
Theorem: (FOSD Property) if and only if for all .
Proof:
If , then for all non-decreasing function . Now by integration by parts, we have . So . Since , we have . Suppose for some we have , since all the probability distributions are right continuous, we can define for and for . This will lead to a contradiction that . It is trivial to show the inverse of this is also true.
Definition: (Second Order Stochastically Domination) Second Order Stochastically Dominate , denoted by , if for all non-decreasing and concave function .
Claim: (SOSD Property) if and only if for all .
Definition: (State) is the State of a finite possible state world.
Definition: (Outcomes) is the set of Outcomes.
Definition: (Anscombe-Aumann Act) An Anscombe-Aumann Act is a function , or we can denote it as . We denote the probability of outcome at state as .
Definition: (Constant Act) A Constant Act is defined as an Anscombe-Aumann Act which satisfies for any .
Definition: (Affine) Function is an Affine if for any , and , we have .
Definition: (Linear) Function is an Affine if for any , and , we have .
Claim: (Mixture Space Theorem) A preference on a convex subset of is complete, transitive, independent and Archimedean if and only if there exists an affine function representing the preference. Moreover, if is an affine representation then is an affine representation of the same preference if and only if there exist and such that .
Claim: (State Dependent Expected Utility) A preference on a convex subset of is complete, transitive, independent and Archimedean if and only if there exists a set of Von-Neumann-Morgenstern utility functions , such that .
Definition: (Changing Act) Given an act , , and a lottery , define a new act .
Definition: (Null State) A state is Null if for any , any we have .
Definition: (State Independence) a preference is state independent if for all non-null states , for all acts , any lottery , implies .
Definition: (Monotonicity) a preference is state independent if for all acts , the constant act implies .
Theorem: (State Independence and Monotonicity) if a preference is state independent, then it is monotonic.
Proof:
Claim: (Expected Utility Theorem) A preference relation on is independent , Archimedean, state independent if and only if there exists a Von-Neumann-Morgenstern utility function , such that , where denote the subjective probability of state .
Definition: (Production Possibility Set) The set , where , and if it is the output, if it is the input.
Axiom: (Properties of Production Possibility Set) The production possibility set satisfies:
Note: Increasing return to scale and additivity don’t imply each other, especially when the other properties do not hold. Consider a production function where irreversibility doesn’t hold.
Theorem: (Properties of Production Possibility Set) the following are true:
Proof:
Definition: (Production Function) Let denotes the outputs and denotes the inputs. Suppose , then define the production function as . The corresponding production possibility set is .
Assumption: (Production Function) The production function is continuous, strictly increasing and strictly quasi-concave on , and .
Definition: (Isoquant) Isoquant is a collection of input combinations which keep output fixed, .
Theorem: (Properties of Production Function) The following are true:
Proof:
Definition: (Separable Production Function) Let be the number of inputs. Suppose we can take partition of , i.e. , then the production function is weakly separable if:
where . Furthermore, if , the production function is strongly separable if:
Definition: (Cost Minimization Problem) The Cost Minimization Problem of the firm is defined as:
Solution: (Cost Minimization Problem)
The solution to this problem is the same as the expenditure minimization problem. The existence of the solution comes from the same theorem. The first order condition are . Combine them with the production constraint we can get the solution.
Claim: (Properties of Cost Function) If satisfies our assumptions, the the cost function has the following properties:
Definition: (Conditional Input Demand) The solution to the cost minimization problem is the conditional input demand function, denoted as .
Claim: (Properties of Conditional Input Demand) Under the assumption of production function, suppose the cost function is twice differentiable, we have
Claim: (Recovering Production Function from Cost Function) For a given function , satisfying properties 1-7 for a cost function, the function is an increasing, unbounded above, quasi-concave function. Moreover, the cost function generated by is .
Claim: (Integrability) If a differentiable function is homogenous of degree 0, and is strictly increasing in , and satisfies symmetry and negative semi-definite of Slutsky matrix, if and only if it is the conditional input demand function generated by some strictly increasing, quasi-concave production function.
Definition: (Profit Maximization Problem) The Profit Maximization Problem of the firm is defined as:
Note: When the production function is constant or increasing return to scale, the solution to the profit maximization problem doesn’t exist.
Solution: (Profit Maximization Problem) The first order conditions are .
Claim: (Properties of Profit Function) If satisfies the assumption, and suppose the profit function exists, then for , we have
Definition: (Input Demand and Output Supply) The solution to the profit maximization problem are the Input Demand and Output Supply functions, i.e. and .
Claim: (Properties of Input and Output) Under the assumption of production function, suppose the cost function is twice differentiable, we have
Definition: (Short Run Cost Minimization Problem) Let the production function be , where is a vector of variable inputs and is a vector of fixed inputs. Then the Short Run Cost Minimization Problem is:
And is called total variable cost, and is called total cost.
Theorem: (MC and AVC) When marginal cost is greater than the average variable cost, i.e. , is increasing, and vice versa.
Proof:
By definition , take differentiation we have . Hence it has the same sign as .
Corollary: (MC and AVC) The MC curve will pass through the minimization point of AVC.
Theorem: (Relationship between Short Run and Long Run) By definition we have . Furthermore, let denote the optimal choice of input at , then we have . This implies that .
Proof:
is true because is the minimization. We have . This implies that minimizes , i.e.. Now take differentiation of the equation , we have .
Definition: (Short Run Profit Maximization Problem) Let the production function be , where is a vector of variable inputs and is a vector of fixed inputs. Then the Short Run Profit Maximization Problem is:
Claim: (Break Even Point) The firm will stop producing when in the short run and it will stop producing when in the long run.
Theorem: (Relationship between Short Run and Long Run) By definition we have . Furthermore, let denote the optimal choice of input at , then we have .
Proof:
is true because is the maximization. We have . This implies that maximize , i.e..
Definition: (Transformation Function) Given a production set the Transformation Function is , such that .
Definition: (Transformation Frontier) Given a production set the Transformation Frontier is , such that .
Definition: (Marginal Rate of Transformation) Given a differentiable transformation function and a point on the frontier, the Marginal Rate of Transformation for good i and j is defined as .
Definition: (Multi Product Firm Profit Maximization) Multi Product Firm Profit Maximization Problem is defined as:
Theorem: (Existence of Profit) If satisfies non-decreasing return to scale, then either or .
Proof:
Suppose at some we have , Since satisfies non-decreasing return to scale, we have for . So is always feasible. As , .
Assumption: (Setup)
Assumption: (Consumers’ Behavior) Agents have complete, transitive, continuous, and strictly convex preference over bundles in .
Definition: (Barter Exchange Economy) An Exchange Economy is defined as:
Definition: (Edgeworth Box) With 2 consumers and 2 goods in the economy, the Edgeworth box is a rectangular diagram with one consumer's origin on one corner, the other's origin on the opposite corner. The width of the box is the total amount of one good, and the height is the total amount of the other good.
Definition: (Contract Curve) With 2 consumers and 2 goods in the economy, the Contract Curve is the subset of allocations where the consumers’ indifference curves through the point are tangent to each other.
Note: Anything on the contract curve and inside the lens is a Barter Exchange Equilibrium.
Definition: (Individual Rationality) A feasible allocation is Individually Rational if for any .
Definition: (Consumer’s Problem) Consumer choose to solve:
Note: Under the normal assumption, this problem has a unique solution given price vector.
Definition: (Excess Demand) Define the excess demand for good k as at price .
Definition: (Aggregate Excess Demand) Define the aggregate excess demand as at price .
Assumption: (Excess Demand) The utility functions are continuous, strictly increasing and strictly quasi-concave on .
Theorem: (Properties of Excess Demand) Under the above assumption about consumer behavior, we have:
Proof:
Continuity follows from continuity of individual demand functions. Homogeneity follows from the individual budget constraint. Now try to prove Walras Law. The budget constraint of each individual can be rewrite as:
Summing over , we get:
If we change the order of summation, we get:
which is what we want to show.
Corollary: (Walras Law) If there are markets in total and markets clear, then the last market clears, too.
Assumption: (Consumer’s Behavior) The utility functions are continuous, strongly increasing and strictly quasi-concave on .
Theorem: (Utility and Aggregate Demand) If each consumer’s utility satisfies the assumption above, and if the aggregate endowment of each good is positive, then the aggregate excess demand satisfies the following properties:
Proof:
Continuity follows from the continuity of individual demands. Walras Law follows from the individual budget constraint. We only need to show the third property.
Suppose is a price vector such that , we need to find a such that property 3 is true. We are going to argue with contradiction. Take a sequence of price vectors , suppose that the aggregate excess demand is bounded, then demand for individual is bounded. Then there is a converging subsequence .
Now construct another allocation such that for and . Then . But since the utility is strongly increasing, we have .
Now because is continuous, there exist a , such that and . Then because as , , there exist a such that and , which is a contradiction to the fact that is the solution to the maximizing problem of the consumer.
Definition: (Walrasian Equilibrium) a vector is called a Walrasian Equilibrium if .
Lemma: (Brower’s Fixed Point Theorem) If is a continuous function mapping from a non-empty, compact and convex subset of to itself, then there is a fixed point such that .
Theorem: (Aggregate Excess and Walrasian Equilibrium) Suppose has the following properties:
Then there is a price vector such that , i.e. the Walrasian Equilibrium exists.
Proof:
For every , let , and let . By definition, is bounded above by 1. Now define:
It’s easy to show that is closed and bounded (hence compact), convex, and non-empty. Now for every good and every , define as follows:
And similarly define . Note that satisfies the following properties:
By Brower’s Fixed Point Theorem, there exists a fixed point . Hence from the definition of , we have:
Hence we can rewrite it as:
which means for every there is a price vector in satisfying the above equation.
Now take a sequence of price vectors that satisfies the above equation, as goes to zero. Since , we have , which is bounded, there exists a subsequence of that converges to a price vector, denoted as .
Note that because its components add up to 1. We claim that . We argue with contradiction. Suppose there is a such that , by property 3 that is given to us in the first place, there is a such that and when goes to zero, is unbounded above.
Then , but , which means the above equation cannot hold, which is a contradiction.
Now we find a price vector . Want to show that .
Note that as , the above equation becomes:
Multiply both sides by and sum over , we get:
by Walras Law (property 2), we have . To summary we have:
Suppose for some , we have , then and hence , which means the above equation cannot hold. Hence for all .
Now remember the Walras Law, we conclude that for all , i.e. .
Corollary: (Existence of Equilibrium) The Walrasian Equilibrium exists.
Assumption: (Setup)
Assumption: (Firm) The firms in the model satisfy the following assumptions:
Definition: (Firm’s Problem) Firm solves the following problem:
Note: Under the assumption above, for any given price vector the solution of the problem is unique, denoted by , which is continuous on and the profit function is well defined.
Definition: (Aggregate Production Possibility) Define the Aggregate Production Possibility Set as .
Note: if , satisfies the assumption above, then so will . Condition 1 and 3 follow directly from the corresponding properties of . The closeness follows too only when boundedness is true.
Theorem: (Aggregate Profit Maximization) For any price , we have if and only if for some , we may write .
Proof:
Assumption: (Consumers’ Behavior) Agents have complete, transitive, continuous, and strictly convex preference over bundles in .
Definition: (Consumer’s Problem) Consumer choose to solve:
Note: If and satisfies the assumptions above, then the solution of the consumer exists and is unique for . Plus, is continuous in and is also continuous.
Definition: (Walrasian Equilibrium) a vector is called a Walrasian Equilibrium if
Theorem: (Existence of The Equilibrium) Consider the economy with and . If All the assumptions are satisfied, and for some production vector , then there exists at least one price vector such that .
Proof:
We will verify that satisfies the three properties of the Equilibrium Existence Theorem.
is continuous because of the continuity of the individual demand function and supply function.
Walras Law is still true because of budget balance.
We want to show that if , and , such that , then with , such that the excess demand of , is unbounded as .
If there is some consumers with strictly positive income at the limit price , then this person’s income will remain to be positive at as due to the fact that is continuous. This person’s demand for good will be unbounded above as .
Hence it suffices to show that there is some consumers with strictly positive income at the limit price .
Because , for some we have . Consider the sum of consumer’s budget constraint:
So there is some person that has strictly positive income, otherwise the above equation would not hold.
Definition: (Feasible Allocation) The Feasible Set is defined as .
Definition: (Pareto Optimality) A feasible allocation is Pareto Efficient if there is no other feasible allocation such that for all , and there exists a such that .
Definition: (Blocking Coalition) Let be a coalition of consumers, We say blocks allocation if there is another allocation such that
Theorem: (Efficiency of Unblocked Allocation) Any unblocked allocation is Pareto Optimal.
Proof:
Suppose not. Then the allocation is not Pareto Optimal, so there is is another feasible allocation such that and such that , which means is blocked by itself.
Note: Not all Pareto Optimal allocations are unblocked.
Definition: (Core) The Core of an exchange economy, denoted as , is the set of all unblocked feasible allocations.
Definition: (Feasible Allocation in Production Economy) An allocation is feasible if .
Definition: (Pareto Optimality in Production Economy) A feasible allocation is Pareto Efficient if there is no other feasible allocation such that for all , and there exists a such that .
Definition: (Blocking Coalition in Production Economy) Let be a coalition of consumers, We say blocks allocation if there is another allocation such that
Definition: (Utility Possibility Set) The Utility Possibility Set is defined as and is feasible, i.e.
Definition: (Pareto Optimality Alternative Definition) A feasible allocation is Pareto Efficient if where .
Definition: (Boundary) Let denote the Boundary of the Utility Possibility Set, also known as the utility frontier: .
Theorem: (Pareto Optimality Alternative Definition) A feasible allocation is Pareto Optimal if and only if .
Proof:
Theorem: (Solution of Pareto Optimality) let the utility functions , for , be continuous. Suppose is strongly increasing, then is Pareto Optimal if and only if it’s a solution to the following problem:
for some .
Proof:
Want to show that the Pareto Optimal allocation solves the problem. By definition, if we set for , then is the maximum utility we could get to give to person 1, hence it’s a solution.
Want to show the solution to the problem is Pareto Optimal. Let be the solution, suppose that is not Pareto Optimal, then there is another allocation, , which is feasible and satisfying for all and there is one such that .
Case 1: when , then automatically is not the solution, contradiction.
Case 2: when for some , and suppose . Without loss of generosity, we could assume the consumption of the first good of consumer is greater than zero, i.e. . Now let be a vector with and for , i.e. . by continuity of , there is a such that .
Now consider another bundle , where , for and . By strongly monotonicity, and is still feasible and everyone else is getting at least as good as . So is not the solution, contradiction.
Case 3: when for some , and suppose . If , then the equation before would become , which is not possible. Since we know , this means the only possible case is . Now we can define a new allocation , where , for and . Then by strongly monotonicity, and is still feasible and everyone else is getting at least as good as . So is not the solution, contradiction.
Definition: (Social Planner’s Problem) Let , consider the following social planner’s problem:
Theorem: (Sufficient Condition for Pareto Optimality) If is a solution to the social planner’s problem, then is Pareto Optimal.
Proof:
Let to be a solution to the social planner’s problem, and suppose is not Pareto Optimal. Then there exists another allocation such that for all , and there exists a such that . Then multiply each side by and add up all the equations, we will get , hence is not the solution to the above problem, Contradiction.
Lemma: (Convexity) Let be concave, then the utility possibility set is convex.
Lemma: (Supporting Hyperplane Theorem) Let be convex, be a point that is not in the interior of , then there exists a vector such that .
Theorem: (Necessary Condition for Social Planner’s Problem) Let be concave, be Pareto Optimal. There exists a vector , such that is a solution to the social planner’s problem.
Proof:
Suppose is Pareto Optimal. Then is on the boundary of the utility possibility set. By Convexity Lemma, is a convex set. By Supporting Hyperplane Theorem, there exists , such that , hence .
It remains to show that . Suppose not, then there is a such that . Then
But this is impossible to hold since can goes to negative infinity. When it does so, and , the right hand side of the equation will goes to positive infinity, but the left hand side stays finite, so we get a contradiction.
Theorem: (Local Non-satiation) Suppose the preference is locally non-satiated, is defined as the solution to the maximizing problem of the consumer given a budget constraint . Then we have:
Proof:
If and suppose , then by local non-satiation there exists another bundle such that and . Hence is not the solution to the maximizing problem, contradiction.
If and suppose , then is not the solution to the maximizing problem, contradiction.
Theorem: (First Welfare Theorem) Suppose that each consumer’s preference are locally non-satiated, then for any allocation that forms a Walrasian Equilibrium with some price vector is Pareto Optimal.
Proof:
Suppose an allocation that forms a Walrasian Equilibrium with some price vector is not Pareto Optimal. Then there is another feasible allocation such that , and . by local non-satiation, we have and . If we combine them, we can get:
Since is the solution to the profit maximization problem of the firm, we have for any . Combine this with the above equation, we have:
which is impossible if the feasible condition holds. Hence we get a contradiction.
Theorem: (Core and Equilibria) Suppose the preference is locally non-satiated, then the Walrasian Equilibrium Allocation is in the core.
Proof:
Suppose not, and let and be a Walrasian Equilibrium Allocation, which is blocked by a subset , then we have for all and there exists such that , and . Since we have local non-satiation, by the Local Non-satiation Theorem, and . Now since is the solution to the profit maximization problem, at price we have . Combine them we can get:
But this cannot hold when feasibility is satisfied. Contradiction.
Definition: (Equilibrium with Transfers) Given an economy , an allocation and a price vector constitute an equilibrium with transfers if there are some wealth levels with where solves the profit maximization problem, and for each , solves the utility maximizing problem with the wealth level assigned to them, and the feasible condition holds, i.e. we have .
Definition: (Income Transfer) Define the Income Transfer of individual as .
Definition: (Quasi-Equilibrium) Given an economy , an allocation and a price vector constitute a Quasi-Equilibrium with transfers if there are some wealth levels with such that:
Note: The Definition of Quasi-Equilibrium will eliminate the problem of boundary solutions. Convexity ensures the existence of a hyperplane that support the better-than set.
Lemma: (Separating Hyperplane Theorem) If and are two disjoint convex subset of , then there exist a vector , such that for all , and for all .
Theorem: (Second Welfare Theorem) Consider an economy , we assume that are convex for all , the preferences are convex and locally non-satiated. for all , then for each Pareto Optimal allocation , there exists a price vector , such that forms a Quasi-Equilibrium with transfers.
Proof:
Aggregation
Start with a Pareto Optimal allocation , define the strictly-better-than sets as , and define . Since we have that preferences are convex, which means for any two bundles and , suppose , then for any , we have , i.e. are convex. Take the sum of finitely many convex set and we will still get a convex set, so is also convex.
Now we aggregate all the firms and define the aggregate production set as , and the set of attainable consumption bundles as . Since are convex, we can conclude that is also convex, and so is .
Separation
Now want to show that . Because is Pareto Optimal, this must be true, otherwise there will exist a in the intersection area such that , which is still feasible, contradicting to the fact that is Pareto Optimal.
Now and are two convex and disjoint set, by the Separating Hyperplane Theorem, there exists a vector , such that for all and for all .
We claim that if for all , then . Take any , by local non-satiation, there exists such that , hence and . So by the separation. Now take a sequence of , then we have , which is what we claimed to be true. Now apply the result to , we have , so by the claim we have .
Similarly, the separation gives us for all , which can be rewrite as . Remember is Pareto Optimal in the first place, and all Pareto Optimal allocations are feasible, i.e. . Multiply by the price, we get .
In a word, we conclude that .
Decentralization
Last we want to show that satisfies the consumer’s conditions of being a part of the Quasi-Equilibrium at price , i.e. we want to show that implies that for some .
Suppose for person we have , define for and then use the claim from step 2, we get that for all , then . Since from step 2 we have , we have the following equation:
Hence .
Similarly, we want to show that for all . Note that following similar steps, we have:
Hence .
In conclusion, forms a Quasi-Equilibrium with transfers.